But a couple of difficulties arise. The first is that if TDT variants can logically separate from each other (i.e. can prove that their decisions aren't linked) then they won't co-operate with each other in Prisoner's Dilemma. We could end up with a bunch of CliqueBots that only co-operate with their exact clones, which is not ideal.

I think this is avoidable. Let's say that there are two TDT programs called Alice and Bob, which are exactly identical except that Alice's source code contains a comment identifying it as Alice, whereas Bob's source code contains a comment identifying it as Bob. Each of them can read their own source code. Suppose that in problem 1, Omega reveals that the source code it used to run the simulation was Alice. Alice has to one-box. But Bob faces a different situation than Alice does, because he can find a difference between his own source code and the one Omega simulated, whereas Alice could not. So Bob can two-box without effecting what Alice would do.

However, if Alice and Bob play the prisoner's dilemma against each other, the situation is much closer to symmetric. Alice faces a player identical to itself except with the "Alice" comment replaced with "Bob", and Bob faces a player identical to itself except with the "Bob" comment replaced with "Alice". Hopefully, their algorithm would compress this information down to "The other player is identical to me, but has a comment difference in its source code", at which point each player would be in an identical situation.

Well, I've had a think about it, and I've concluded that it would matter how great the difference between TDT and TDT-prime is. If TDT-prime is almost the same as TDT, but has an extra stage in its algorithm in which it converts all dollar amounts to yen, it should still be able to prove that it is isomorphic to Omega's simulation, and therefore will not be able to take advantage of "logical separation".

But if TDT-prime is different in a way that makes it non-isomorphic, i.e. it sometimes gives a different output given the same inputs, that may still not be enough to "separate" them. If TDT-prime acts the same as TDT, except when there is a walrus in the vicinity, in which case it tries to train the walrus to fight crime, it is still the case in this walrus-free problem that it makes exactly the same choice as the simulation (?). It's as if you need the ability to prove that two agents necessarily give the same output for the particular problem you're faced with, without proving what output those agents actually give, and that sure looks crazy-hard.

EDIT: I mean crazy-hard for the general case, but much, much easier for all the cases where the two agents are actually the same.

EDIT 2: On the subject of fairness, my first thoughts: A fair problem is one in which if you had arrived at your decision by a coin flip (which is as transparently predictable as your actual decision process - i.e. Omega can predict whether it's going to come down heads or tails with perfect accuracy), you would be rewarded or punished no more or less than you would be using your actual decision algorithm (and this applies to every available option).

EDIT 3: Sorry to go on like this, but I've just realised that won't work in situations where some other agent bases their decision on whether you're predicting what their decision will be, i.e. Prisoner's Dilemma.

The right place to introduce the separation is not in between TDT and TDT-prime, but in between TDT-prime's output and TDT-prime's decision. If its output is a strategy, rather than a number of boxes, then that strategy can include a byte-by-byte comparison; and if TDT and TDT-prime both do it that way, then they both win as much as possible.